You prune first. I would wait till fall to transplant, but prune now. So, you prune both the roots(cut around the bush with a shovel) and prune the above ground part. Pruning the roots allow the plant to put out new roots in the transplanted root ball and pruning the above ground reduces the demand placed in the roots to supply water. The plant only takes water in through the very fine roots. When you transplant you will lose 40-75 % of these roots.

Don't forget to add fertilizer when you prune them back. I got some at Betty's Azalea ranch. But I'm sure there are lots of places to get the fertilizer.

It will be easier on your husband and the plants if you prune first, then move. Your bushes will be less stressed after you move them if they have less foliage to lose water through while they're trying to regrow a lot of their roots.

Move, then prune. Or, just buy new ones. We cut ours down to the stub trying to kill them and they grew back quickly to full plant and look really nice (well, the neighbors like them and they are on the side of the house we can't see so I left them).

We have some azaleas that are flourishing in the wrong part of the yard. They are also getting kind of overgrown and leggy. Now that all the blooms are gone, I want to severely prune them back - but we also want to move them. Which would you do first?
I am inclined to prune first, but only because I am lazy and can do that here and there, and make DH do the actual moving